Volume VIII — Electronic Structure and Many-Body Matter#
Volume VII ended with quantum statistics and, in its Coda, the language of many bodies; this volume is where that language earns its keep. Everything before it treated the quantum mechanics of one particle, or the statistics of many that ignore each other. But the matter on every desk — the silicon in the machine running this page, the metal in its frame — is neither: it is \(10^{23}\) electrons interacting through the Coulomb force, and the single problem “solve the many-electron Schrödinger equation” has consumed more computer cycles than any other question science asks. This volume is about the ideas that make that problem tractable, built and tested one approximation at a time.
An exact laboratory, then honest approximations#
The volume’s method is its signature. Early on we construct a system small enough to solve exactly — two interacting electrons on a grid — and from then on every approximation the field actually uses is judged against exact numbers we computed ourselves. Hartree–Fock is derived, coded, and measured against the exact answer; its error acquires a name (correlation) and a number. Density-functional theory is built from the Hohenberg–Kohn theorems up through a working Kohn–Sham code, and the exact exchange-correlation potential — usually a phrase in a review article — is computed outright by inverting the exact density. When the local-density approximation errs, we do not report that it errs: we measure by how much, and we trace the failure to the exact conditions it violates. The band-gap problem, the self-interaction error, the derivative discontinuity: each is a computation here, not a warning.
Four movements#
The many-electron problem (8.1–8.4) states the problem honestly — the Born–Oppenheimer separation, the wall of exponential cost, the exact laboratory — and takes mean-field theory as far as it goes, from Hartree–Fock atoms to the homogeneous electron gas whose exchange energy will power everything after.
Density-functional theory (8.5–8.8) is the reigning answer: Thomas–Fermi as the prototype, the Hohenberg–Kohn theorems run as computations, the Kohn–Sham construction as a working radial code, and the exact conditions — piecewise linearity, the derivative discontinuity — that separate the exact functional from the approximations in daily use.
Electrons in crystals (8.9–8.12) moves to solids: tight binding through graphene’s Dirac cones, plane waves and pseudopotentials, the empirical pseudopotential band structures of real silicon and gallium arsenide, and the geometry of Bloch states — Berry phases, Wannier functions, and the SSH model’s topological edge states — that a modern electronic-structure course cannot omit.
Beyond the mean field (8.13–8.17) is where the volume earns its subtitle: the Hubbard model diagonalized exactly, spectral functions and the GW approximation tested against exact answers, optical absorption and excitons, time-dependent DFT propagated in real time against the exact laboratory, and — closing the volume and the course’s physics — the BCS theory of superconductivity, electrons doing something no single electron can.
A coda follows the four movements, and it is deliberately retrospective. Every notebook above represents a wavefunction one of two ways: on a real-space grid, or in plane waves. Neither is what quantum chemistry actually uses. §8.18 arrives at the third representation — Gaussians centred on the atoms — only after both, which is the honest order, because the contrast is the lesson. It is also the only representation in which a particular pathology can exist at all: a basis attached to atoms rather than to space lends its functions to its neighbours, and the energy it returns is lower than it has any right to be. The whole coda runs on a single electron, where every integral is a closed form, and it ends by measuring an error the plane waves of §8.10 are structurally incapable of making.
Provenance#
This volume follows the shape of the electronic-structure education I received at Humboldt-Universität zu Berlin: Claudia Draxl’s density-functional theory course, and Pasquale Pavone’s electronic-structure theory and theoretical solid-state physics courses, whose problem sets I solved as a student and whose spirit — theory stated precisely, then made concrete — these notebooks try to honour. The exercises here are original, written for this course’s computational format, but their themes descend from those courses and from the standard literature: Martin’s Electronic Structure, Parr and Yang, Giustino’s Materials Modelling using Density-Functional Theory, and the primary papers cited notebook by notebook. The worked literature example of the fourth movement — quasiparticle and excitonic corrections for the photocathode materials Na₂KSb and NaK₂Sb — is my own MSc thesis (Humboldt-Universität zu Berlin, 2019, supervised by Caterina Cocchi).
Readers who want the production-code counterpart of this volume — the same physics driven through CP2K and Quantum ESPRESSO rather than built from scratch — will find it in the companion course Molecular and Materials Modelling; the two are cross-referenced where they meet, most directly at the silicon band structure.